My confusion arises in determining the derivatives. In a curved spacetime, a vector exists in the tangent plane to a point. This point is labeled by some coordinates and so when we calculate e.g. $\frac{\partial r}{\partial x} = \sin \theta \cos \theta$ we can plug in the coordinates $\theta, \phi$ and hey presto. But in a euclidean space, the tangent space is the manifold. It doses not seem to make sense that in order to transform between coordinate systems we need both the 'location' of the vector at the coordinates, and the components of the vector itself?

1 Answer
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I think this is where your confusion lies. A flat Euclidean manifold $M$ still has a distinct tangent space $T_P$ at each point $P$, and a vector at $P$ lives in the tangent space $T_P$. Each tangent space happens to resemble $M$ iself but this is just coincidence. The vectors in a vector space on $M$ may be informally represented as if they lived in $M$ itself, but this is a simplified and somewhat misleading picture.

$\begingroup$Hmmm ok I can accept that. But still, if we have some vector located at the origin (0,0,0), but with non-zero components in the Cartesian basis ($A^x, A^y, A^z$) then how would one do the transform to the spherical polar basis?$\endgroup$
– user1887919Jul 30 '19 at 10:47

$\begingroup$@user1887919 (0,0,0) is not a good point for spherical polar coordinates. Your map is not a diffeomorphism (at the point).$\endgroup$
– lalalaJul 30 '19 at 10:56